Find a term, the common difference, or the sum of a sequence that changes by a constant amount each step — every formula here builds off just two numbers: the first term and the common difference. Every question comes with a written walkthrough of which formula applied.
The first term uses n=1, which makes (n−1)=0 — plugging n directly into the formula without subtracting 1 shifts every term by one position.
If a sequence is described starting from its 5th term, that term isn't a₁ — the nth term formula still needs the actual first term, or has to be adjusted to count from a different index.
Sₙ = n/2(a₁+aₙ) needs the first and last term specifically, not the first and second, or any other pair — using the wrong pair changes the average being scaled by n.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Find the 10th term of the arithmetic sequence 4, 9, 14, 19, ...
A — 49. Common difference d=5, first term a₁=4: a₁₀ = 4 + (10−1)(5) = 4+45 = 49. 45 (B) comes from computing (n−1)d correctly but forgetting to add the first term a₁ back in.
In an arithmetic sequence, the 3rd term is 11 and the 7th term is 27. Find the first term.
A — 3. Set up two equations from the given terms: a₁+2d=11 and a₁+6d=27. Subtracting gives 4d=16, so d=4, and a₁=11−2(4)=3. -1 (B) comes from using (n) instead of (n−1) as the exponent index — solving as if the 3rd term meant a₁+3d instead of a₁+2d, shifting every calculation by one position.
The sum of the first 20 terms of an arithmetic sequence is 670. If the first term is 5, find the common difference.
A — 3. Use the sum formula: S₂₀ = 20/2(a₁+a₂₀) = 10(5+a₂₀) = 670, so a₂₀ = 62. Then apply the nth term formula: a₂₀ = a₁+19d, so 5+19d=62, giving 19d=57 and d=3. 2.85 (B) comes from dividing 57 by 20 instead of 19 — using n instead of (n−1) in the final division.
Arithmetic sequence problems reward tracking exactly which term index you're on — one off-by-one slip in (n−1) shifts every downstream calculation.
Label the three pieces you have before choosing a formula, and double-check whether a given term is really a₁ or a later term. Untimed practice is where that labeling habit sticks.
Move to timed sessions once term-index tracking is automatic. Problems that require solving a system for a₁ and d run longer than direct term lookups.
Pair arithmetic sequences with geometric sequences in a mock — the setup logic is nearly identical, just addition versus multiplication.
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